Changeset 2834:4eed73fe6f31

Timestamp:

02/07/07 20:48:06 (6 years ago)

Author:

William Stein <wstein@…>

Branch:

default

Message:

add examples.

File:

• sage/schemes/elliptic_curves/ell_rational_field.py (modified) (4 diffs)

sage/schemes/elliptic_curves/ell_rational_field.py

@@ -3197 +3197 @@
             misc.verbose("index = %s"%ind)
             # Compute upper bound on square root of index.
-            if ind.length() < 1:
+            if ind.absolute_diameter() < 1:
                 t, i = ind.is_int()
                 if t:   # unique integer in interval, so we've found exact index squared.
@@ -3224 +3224 @@
                            ignore_nonsurj_hypothesis=False):
         """
-        Given a fundamental discriminant D (=-3,-4) that satisfies the
+        Given a fundamental discriminant D (!= -3,-4) that satisfies the
         Heegner hypothesis, return a list of primes so that
         Kolyvagin's theorem (as in Gross's paper) implies that any
@@ -3277 +3277 @@
                 correct up to addition or a real number with absolute
                 value less than $10^{-10}$.
+
+        EXAMPLES:
+            sage: E = EllipticCurve('37a')
+            sage: E.shabound_kolyvagin()
+            ([2], 1)
+            sage: E = EllipticCurve('141a')
+            sage: E.sha_an()
+            1
+            sage: E.shabound_kolyvagin()
+            ([2, 7], 49)
+
+        We get no information the curve has rank $2$.
+            sage: E = EllipticCurve('389a')
+            sage: E.shabound_kolyvagin()
+            (0, 0)
+            sage: E = EllipticCurve('681b')
+            sage: E.sha_an()
+            9
+            sage: E.shabound_kolyvagin()
+            ([2, 3], 9)        
+        
         """
         if self.has_cm():
@@ -3341 +3362 @@
             if t:
                 break
-            elif I.length() < 1:
+            elif I.absolute_diameter() < 1:
                 raise RuntimeError, "Problem in shabound_kolyvagin; square of index is not an integer -- D=%s, I=%s."%(D,I)
             misc.verbose("Doubling bounds")